Spectra of a class of self-affine measures
A probability measure \(\mu\) is denoted by of compact support on \(\mathbb{R}^n\) is called a spectral measure if there exists a discrete set \(\Lambda\subset\mathbb{R}^n\) such that \(E(\Lambda):=\{e^{2\pi i(\lambda,x)}:\lambda\in\Lambda\}\) forms an orthogonal basis for \(L^2(\mu)\); the set \(\Lambda\) is then called a spectrum for \(\mu\) and \((\mu,\Lambda)\) is called a spectral pair; the iterated function system (IFS) is \(\{\Phi_d(x)= M^{-1}(x+ d)\}_{d\in D}\), where \(M\in M_n(\mathbb{Z})\) is an expanding integer matrix, that is, all the eigenvalues of the integer matrix have modulus greater than 1 and \(D\subset\mathbb{Z}^n\) is a finite subset whose cardinality is denoted by \(|D|\). The self-affine measure considered here is a unique probability measure \(\mu:= \mu_{M,D}\) satisfying \[ \mu= {1\over|D|} \sum_{d\in D}\mu\circ \Phi^{-1}_d, \] supported on the invariant set \(T(M,D)\) of IFS \(\{\Phi_d(x)\}_{d\in D}\), where \[ T(M,D)= \Biggl\{\sum^\infty_{j=1} M^{-j} d_j: d_j\in D\Biggr\}. \] For two finite subsets \(B\) and \(S\) of \(\mathbb{R}^n\), of the same cardinality \(q\), we say that \((B,S)\) is a compatible pair if the \(q\times q\) matrix \(H_{B,S}:= [q^{{-1\over 2}} e^{2\pi i(b,s)}]_{b\in B,s\in S}\) is unitary. There are several questions on the spectrality of a self-affine measure \(\mu_{M,D}\) and its relation with a compatible pair. The author discusses some questions and answers one such question posed by Dutkey and Jørgensen. Further, he investigates certain spectral properties of Bernoulli convolutions \(\mu_\lambda\) which are a class of self-affine fractal measures on invariant sets \(K_\lambda\) arising from LFSs \[ \{\tau_+(x)= \lambda(x+ 1), \tau_-(x)= \lambda(x- 1)\} \] on \(\mathbb{R}\), where \(\lambda\in(0, 1)\). For a spectral self-affine measure \(\mu_{M,D}\), it is an interesting topic to characterize all its spectra. Under certain conditions, it is shown that the integer spectrum of a spectral self-affine measure \(\mu_{M,D}\) has the following spectral property: with the expanding matrix \(M\in M_n(2)\), \(D\) and \(S\) two finite subsets of \(\mathbb{Z}^n\) of the same cardinality such that \((M^{-1}D,S)\) is a compatible pair and \(0\in D\cap S\), if \(\Lambda_1\subseteq\mathbb{Z}^n\) is a spectrum for \(\mu_{M,D}\), then \(M^*\Lambda_1+ S\subseteq\mathbb{Z}^n\) is also a spectrum for \(\mu_{M,D}\)
scientific article; zbMATH DE number 5863603
scientific article